On the question of second sound in germanium: a theoretical viewpoint

We revisit the recent experimental work from Beardo<i>et al</i>(2021<i>Sci. Adv.</i><b>7</b>eabg4677) wherein the observation of second sound in germanium is claimed. Through the lens of the phonon Boltzmann transport equation, we seek theoretical evidence of second sound. We begin with direct solutions to the linearized phonon Boltzmann transport equation (LBTE) for a frequency modulated heat source which do not reveal the presence of second sound. We then review the requirements imposed on the collision operator (or equivalently, the full scattering matrix) of the LBTE for the observation of driftless second sound as established by Hardy. By performing an eigendecomposition of the full scattering matrix, we show that the requirement that the smallest nonzero eigenvalue must be associated with an odd eigenvector is not satisfied. Finally, numerical solutions to the BTE under the relaxation time approximation in the 1D frequency-domain thermoreflectance experimental geometry demonstrate that phase lag alone is not a suitable experimental observable for inferring second sound. We conclude by suggesting alternative explanations that may explain the experimental data and discussing the need for a second sound 'smoking gun'.